Generation Of Transient Waves By Impulsive Disturbances In An Inviscid Fluid With An Ice-cover

Arch Appl Mech (2006) 76: 49–63 DOI 10.1007/s00419-006-0004-1

O R I G I NA L

D. Q. Lu · S. Q. Dai

Generation of transient waves by impulsive disturbances in an inviscid fluid with an ice-cover

Received: 12 September 2005 / Accepted: 9 December 2005 / Published online: 31 January 2006 © Springer-Verlag 2006

Abstract The dynamic responses of an ice-covered fluid to impulsive disturbances are analytically investigated for two- and three-dimensional cases. The initially quiescent fluid of infinite depth is assumed to be inviscid, incompressible and homogenous. The thin ice-cover is modelled as a homogenous elastic plate with negligible inertia. Four types of impulsive concentrated disturbances are considered, namely an instantaneous mass source immersed in the fluid, an instantaneously dynamic load on the plate, an initial impulse on the surface of the fluid, and an initial displacement of the ice plate. The linearized initial-boundary-value problem is formulated within the framework of potential flow. The solutions in integral form for the vertical deflexions at the ice-water interface are obtained by means of a joint Laplace-Fourier transform. The asymptotic representations of the wave motions for large time with a fixed distance-to-time ratio are derived by making use of the method of stationary phase. It is found that there exists a minimal group velocity and the wave system observed depends on the moving speed of the observer. For an observer moving with the speed larger than the minimal group velocity, there exist two trains of waves, namely the long gravity waves and the short flexural waves, the latter riding on the former. Moreover, the deflexions of the ice-plate for an observer moving with a speed near the minimal group velocity are expressed in terms of the Airy functions. The effects of the presence of an ice-cover on the resultant wave amplitudes, the wavelengths and periods are discussed in detail. The explicit expressions for the free-surface gravity waves can readily be recovered by the present results as the thickness of ice-plate tends to zero. Keywords Waves · Ice-cover · Impulsive disturbances · Asymptotic · Group velocity 1 Introduction The dynamics of an ice-covered fluid due to moving loads on the ice sheets or on the fluid has been investigated by many researchers in view of its practical importance and theoretical interest [1–6]. A contemporary review on this subject was provided by Squire et al. [1] and the historic references were given therein. Recently, Miles and Sneyd [2] developed a numerical method for the deflection of an ice-sheet due to a line load moving at varying speed. Chowdhury and Mandal [3] analytically derived the velocity potentials for the motion generated by a ring source in an ice-covered fluid. Mandal and Basu [4] studied the wave diffraction due to a small elevation of the bottom of an ice-covered ocean. Maiti and Mandal [5,6] analytically investigated the generation of waves by two kinds of two- and three-dimensional initial disturbances acting on the surface of water with an ice-cover and derived the asymptotic representations of the wave motions for large time with a fixed distance-to-time ratio. Thus, the classical Cauchy-Poisson problems, which are concerned with the wave motions due to disturbances originating at the free surface [7, Sect. 6.4], were generali´zed by Maiti and D.Q. Lu (B)· S.Q. Dai Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Yanchang Road, Shanghai 200072, China. E-mail: [email protected], [email protected]

50

D.Q. Lu, S.Q. Dai

Mandal [5,6] who replaced the free surface with an ice-cover. Actually, there have been several attempts to extend the Cauchy-Poisson wave problems to complicated physical configurations, which are due to Miles [8] and Debnath [9] who took the effect of viscosity into account, Chen and Duan [10] who considered the effect of surface tension, Lu and Chwang [11] who introduced the laminar interaction of a Stokeslet with a free surface, and Lu et al. [12] who studied the cases of two semi-infinite inviscid fluids. Using an asymptotic analysis and graphic illustration, Maiti and Mandal [5,6] made a comparison between the wave motions in the ice plate and the free-surface gravity waves in the fluid for two- and three-dimensional cases. However, it is felt that the general characteristics of the wave responses to disturbances acting on the ice-covered fluid have not been fully elucidated. In this paper, dynamic responses of an ice-covered fluid to impulsive disturbances will be considered. The fluid is initially quiescent, infinitely deep and is assumed to be inviscid, incompressible and homogenous. The ice-cover is assumed to be thin enough to be modelled as a homogenous elastic plate with negligible inertia. There are four types of impulsive concentrated disturbances to be considered, namely an instantaneous mass source immersed in the fluid, an instantaneously dynamic load on the plate, an initial impulse on the surface of the fluid, and an initial displacement of the ice-sheet. Within the regime of linear theory, any spatially distributed and/or temporally prescribed disturbances can be constructed by a superposition of the four fundamental concentrated disturbances, and the type of fundamental ones to be used depends on the physical nature of the problem under consideration. In Sect. 2, the general mathematical model is formulated for two- and threedimensional cases. Four types of disturbances are simultaneously included in the mathematical formulation. In Sect. 3, based on the formal solutions for the displacement of the ice-water interface due to four types of two-dimensional disturbances, the asymptotic representations for the generated waves are derived for large time with a fixed distance-to-time ratio. In Sect. 4, the solutions for the three-dimensional cases are provided. Finally, discussion and conclusions are made in Sect. 5. 2 General mathematical formulation Without loss of generality, a Cartesian coordinate is used in which the z axis points vertically upward while z = 0 represents the mean ice-water interface. Therefore, the governing equation is ∇ 2
= Mδ(x − x0 )δ(t − t0 ),

(1)

where
(x, t; x0 , t0 ) is the velocity potential for the perturbed flow, M the constant strength of the simple source, δ(·) the Dirac delta function, x an observation point, t the time, x0 the source point and t0 the instant at which the source is applied. For two-dimensional cases, x = (x, z) and x0 = (x 0 , z 0 ) while for three-dimensional cases, x = (x, y, z) and x0 = (x 0 , y0 , z 0 ). It is assumed that the wave amplitude is very small in comparison with the wavelength. Thus, the linearized boundary conditions will be applied on the undisturbed ice–water interface. The kinematic and dynamic conditions at z = 0 are given by ∂ζ ∂
− = 0, ∂t ∂z ρ

∂ 2ζ ∂
+ ρgζ + D∇ 4 ζ + ρe h 2 = −Pδ(z − z0 )δ(t − t0 ), ∂t ∂t

(2)

(3)

where ζ is the vertical deflexion of the ice-water interface; ρ and ρe are the uniform densities of the fluid and the plate, respectively; g is the acceleration of gravity; D = Eh 3 /[12(1 − ν 2 )] is the flexural rigidity of the plate; E, h and ν are the Young’s modulus, the thickness and the Poisson’s ratio of the plate, respectively; P is the constant strength of the applied load; z and z0 are the field point and the source point at z = 0, respectively. For two-dimensional cases, z = (x, 0) and z0 = (x 0 , 0) while for three-dimensional cases, z = (x, y, 0) and z0 = (x 0 , y0 , 0). Equation (2) implies that there is no cavitation between the plate and the water surface and fluid particles once in between will always remain there. According to the formulation depicted by Squire et al. [1, Eq. (3.44)], Eq. (3) states that the elastic plate is subject to an impulsive downward concentrated load. Historically, there have been two approaches available to pose initial conditions for the problem considered here. The first one involves the initial values of
and its first time derivative, as was formulated by Stoker [7, Eqs. (6.1.7) and (6.7.4)]. Another involves the initial values of
and ζ , as was proposed by Miles [13,

Generation of transient waves by impulsive disturbances in an inviscid fluid with an ice-cover

51

Eq. (1.6)] and Mei [14, Sect. 2.1]. It is noted that Stoker’s initial condition is regarded by Miles [13] as a “ spurious” one from the physical point of view. As it is, these two approaches are equivalent if and only if the pressure at the instant t = 0 equals to zero, as was stated by Miles [13]. Eliminating ζ from Eqs. (2) and (3), one readily gets the combined boundary condition similar to the result provided by Maiti and Mandal [5, Eq. (2.4); 6, Eq. (4)] who imposed the initial values on
+

ρe h ∂
ρ ∂z

in our notation and its first time derivative. In view of Miles’s idea [13, Eq. (1.6); 2, Eq. (2.3)], we impose the initial conditions on
, ζ and the first time derivative of ζ in order to meet the requirement for Eqs. (2) and (3) to be a well-posed initial-boundary-value problem. Therefore, the initial conditions at z = 0 are
|t=0 = −

I0 δ(z − z0 ), ρ

ζ |t=0 = E 0 δ(z − z0 ),
∂ζ

= 0. ∂t
t=0

(4) (5) (6)

Equations (4) and (5) imply that the fluid is initially subject to a surface impulse and elevation concentrated at z0 , where I0 and E 0 are the corresponding constant magnitudes. Equation (6) represents the initial vertical velocity of the ice-water interface is zero, which is consistent with the assumption that the entire fluid is at rest for t < 0. Moreover, since the finite disturbance must die out at infinity, it is required that ∇
→ 0 as z → ∞,

(7)

which imposes a uniqueness on the problem considered. Thus, Eqs. (1)–(7) constitute a well-posed problem for
and ζ , mathematically as well as physically. It should be noted that the physical units for M in the two and three dimensions are different, so are P, I0 and E 0 . However, the two- and three-dimensional problems have the same analytic forms, as shown in Eqs. (1)–(7). The assumption of linearity allows us to envisage the perturbed flow as the superposition of a singular flow and a regular flow. The former represents the effect of a moving singularity in an unbounded domain while the latter the influence of the boundary. Thus, we write
=
S (x, t; x0 , t0 ) +
R (x, t),

(8)

where
S is the potential due to the simple source while
R is a continuous function everywhere in the corresponding domain. For the regular component in Eq. (8), we have ∇ 2
R = 0.

(9)

Thus, the relation between the singular and regular components can be established through the boundary conditions at z = 0: ∂ζ ∂
R ∂
S − = , ∂t ∂z ∂z ρ

∂ 2ζ ∂
S ∂
R + ρgζ + D∇ 4 ζ + ρe h 2 = −Pδ(x − x0 )δ(t − t0 ) − ρ . ∂t ∂t ∂t

(10) (11)

3 Formal solution and asymptotic representation for the two-dimensional problem For two-dimensional case, the singular component in Eq. (8) is the fundamental solution of Laplace equation in an unbounded domain,   M 1 S (12) ln δ(t − t0 ),
=− 2π r where r = ||x − x0 ||. By taking the Laplace-Fourier transform over Eq. (1) and applying the Jordan lemma and the Cauchy residue theorem, an alternative representation for Eq. (12) can be given by

52

D.Q. Lu, S.Q. Dai

M
=− 2 8π i

c+i∞ 

+∞ 1 ds dα exp[−k|z − z 0 | + i f + s(t − t0 )], k

S

(13)

−∞

c−i∞

where k = |α|, f = α(x − x 0 ), c is the Laplace convergence abscissa. In order to obtain the formal solution of this initial-boundary-value problem, it is convenient to introduce a combination of the Laplace transform with respect to t and a Fourier transform with respect to spatial variables. For two-dimensional cases, we have c+i∞ 

+∞ ˜ R exp(kz), ζ˜ } exp(i f + st). ds dα {


1 {
, ζ } = 4π 2 i R

c−i∞

(14)

−∞

By substituting Eqs. (13) and (14) into the Laplace-Fourier transforms of boundary conditions (10) and (11), two ˜ R and ζ˜ , which can readily be solved. simultaneous algebraic equations are set up for the unknown functions
Consequently, the formal integral expression for the displacement of ice-water interface can be written as ζ = Mζ1 (x, t; x0 , t0 ) +

P I0 ζ2 (z, t; z0 , t0 ) + ζ3 (z, t; z0 ) + E 0 ζ4 (z, t; z0 ), ρ ρ

(15)

where 1 {ζ1 , ζ2 , ζ3 , ζ4 } = 2π

+∞ −∞

exp(i f ) dα 1 + σk

×{exp(kz 0 ) cos(ωt − ωt0 ), −  ω(k, h) =

k k sin(ωt − ωt0 ), − sin(ωt), (1 + σ k) cos(ωt)}, (16) ω ω

1 + γ k4 1 + σk

1/2 ω0 ,

(17)

√ and ω0 = ω(k, 0) = gk, γ = D/ρg, σ = hρe /ρ. Equation (17) is known as the dispersion relation. A special case of Eq. (17) with h = 0 corresponds to the dispersion relation for the classical Cauchy-Poisson wave problems. In the cases of general disturbances formulated by continuously distributed functions M(x, t), P(z, t), I0 (z) and E 0 (z), one can, at least theoretically, construct the resultant displacement of the ice-water interface as   1 ζ (z, t) = M(x0 , t0 )ζ1 (z, t; x0 , t0 )dx0 dt0 + P(z0 , t0 )ζ2 (z, t; z0 , t0 )dz0 dt0 ρ P SM   1 + I0 (z0 )ζ3 (z, t; z0 )dz0 + E 0 (z0 )ζ4 (z, t; z0 )dz0 , (18) ρ I

E

where S M ∈ 2 × [0, +∞); P ∈  × [0, +∞); I , E ∈  are corresponding domains of the distributed functions. Without loss of generality, we set x0 = y0 = t0 = 0 hereinafter for the two- and three-dimensional problems. Thus, ζ2 in Eq. (16) is identical with ζ3 and will not be shown next. Furthermore, we re-write Eq. (16) as +∞   2 2  ik 1  exp(it mn ) dk {ζ1 , ζ3 , ζ4 } = exp(kz 0 ), (−1)n+1 , 1 + σ k , 4π 1 + σk ω

(19)

m=1 n=1 0

where

mn (k) = (−1)m+1 k

x + (−1)n+1 ω. t

(20)

Generation of transient waves by impulsive disturbances in an inviscid fluid with an ice-cover

53

It is noted that ζ4 in Eq. (19) agrees with Maiti and Mandal’s Eq. (3.10) [5] for the deflexion due to a displacement concentrated at the origin. The physical characteristics of the wave motion are not explicitly seen in the integral representation (19). Moreover, the accurate evaluation of this integral is extremely difficult, and in general can only be performed numerically. In order to obtain the principal physical features of the wave motion, it is necessary to adopt the asymptotic analysis for the displacement of the ice-sheets. Next, the asymptotic behavior of Eq. (19) shall be studied for large t with a fixed distance-to-time ratio. The method of stationary phase is used for the k integration in Eq. (19). This approximation has been successfully used to study the classical Cauchy-Poisson problem [9,13]. According to the stationary-phase approximation, the dominant contribution to the integral in Eq. (19) stems from the stationary points of mn . It is easily seen that for x > 0, 12 and 21 have stationary points while for x < 0, 22 and 11 have stationary points, and the stationary points for both x > 0 and x < 0 are the same. The solutions for the stationary points, denoted by k j , are determined by ∂ mn = 0. ∂k

(21)

A straightforward derivation for Eq. (21) yields |x| − Cg t  √ g 1 + 5γ k 4 + 4γ σ k 5 1 1 = 0, = √ −√ · 2 k0 k (1 + σ k)3/2 (1 + γ k 4 )1/2

Q(k, k0 , h) =

(22)

where C g (k, h) = ∂ω/∂k is the group velocity, and k0 = gt 2 /4x 2 . It is noted that k0 is the special root of Q(k, k0 , 0) = 0, which corresponds to the two-dimensional Cauchy-Poisson gravity waves. Generally speaking, the explicit analytical solutions for Eq. (22) cannot readily be given for arbitrary h, γ , σ , and |x|/t. However, once the physical parameters h, γ and σ are given, the nature of roots with respect to k for Eq. (22) depends on the value of |x|/t only. To have a graphical representation for the theoretical results, we adopt hereinafter physical parameters given by Squire et al. [1, p. 105], E = 5 GPa, ν = 0.3, ρ = 1024 kg m−3 , ρe = 917 k gm−3 and g = 9.8 ms−2 . Figure 1 shows curves for the group velocities C g (k, h). It can be seen from Fig. 1 that for a given h, there exists a minimal group velocity, denoted by C gmin (h) = C g (kc , h), at which Eq. (22) has one real positive root kc (h) only and ωc = ∂ 2 ω(kc , h)/∂k 2 = 0. As h increases, C gmin increases while kc decreases. When |x|/t > C gmin , Eq. (22) has two real positive roots, k1 (|x|/t, h) and k2 (|x|/t, h) with 0 < k1 < k2 < +∞. For a fixed |x|/t > C gmin , k1 decreases slowly from k0 as h increases from zero while k2 decreases rapidly from the infinity.

Cg

10

(vi) (v) (iv)

(iii)

5 (ii)

Cg= x /t Cg min

(i)

0 0

kc

5

10

15

k

Fig. 1 Group velocity curves C g (k, h) with (i) h = 0 m, (ii) h = 0.001 m, (iii) h = 0.005 m, (iv) h = 0.01 m, (v) h = 0.05 m, and (vi) h = 0.1 m

54

D.Q. Lu, S.Q. Dai

When |x|/t > C gmin , according to the standard stationary-phase approximation [15, Sect. 3.4], the expansion for the phase function near k j is taken as

mn (k) ≈ mn (k j ) +

1 ∂ 2 mn (k j ) (k − k j )2 . 2 ∂k 2

(23)

By a straightforward application of the method of stationary phase, the asymptotic representation of Eq. (19) can be given as 2  [ζ1 j , ζ3 j , ζ4 j ], {ζ1 , ζ3 , ζ4 }

(24)

j=1

where {ζ1 j , ζ3 j , ζ4 j } =



1

exp(k j z 0 ) cos ψ j ,

(2π|ωj |t)1/2 (1 + σ k j )

 kj sin ψ j , (1 + σ k j ) cos ψ j , ωj

π ψ j = k j |x| − ω j t − sgn(x)sgn(ωj ) , 4

(25)

(26)

ω j = ω(k j , h), and ωj = ∂ 2 ω(k j , h)/∂k 2 , sgn(x) = ±1 as x ≷ 0. It should be noted that Eq. (25) holds for ωj = 0 only. As |x|/t decreases to C gmin , k1 and k2 move together toward kc while ωj tends to zero. Accordingly, Eq. 3 3 (25) predicts an infinitely increasing wave amplitude. It is noted that ω j = ∂ ω(k j , h)/∂k = 0. To have a better approximation for Eq. (19) when k j is sufficiently close to kc , the expansion for the phase function is taken as

mn (k) ≈ mn (k j ) +

1 ∂ 2 mn (k j ) 1 ∂ 3 mn (k j ) 2 (k − k ) + (k − k j )3 . j 2 ∂k 2 6 ∂k 3

(27)

Furthermore, according to Scorer [16], Eq. (19) can be approximately expressed as {ζ1 , ζ3 , ζ4 }

2  2 

1

1/3 (1 + σ k ) (32ω j j t) m=1 j=1



×Re where

exp(iψmI j )

  I m ik j I I exp(k j z 0 )Ki(Z j ), (−1) Ki(Z j ), (1 + σ k j )Ki(Z j ) , ωj 

ψmI j = (−1)m+1 k j |x| − ω j t −

Z Ij = −

(ωj )2 t 2/3 4/3 22/3 (ω j )

1 Ki(Z ) = Ai(Z ) + iGi(Z ) = π

∞

(ωj )3 t 2 3(ω j )

(28)

,

,

  u3 exp i u Z + du, 3

(29)

(30)

(31)

0

and Re(Z ) stands for the real part of Z , Ai(Z ) is the Airy integral and Gi(Z ) the conjugate integral [17, p. 25]. When |x|/t = C gmin , ∂ mn (kc )/∂k = ∂ 2 mn (kc )/∂k 2 = 0 and ωc = ∂ 3 ω(kc )/∂k 3 = 0. In this case, the expansion for the phase function is taken as

mn (k) ≈ mn (kc ) +

1 ∂ 3 mn (kc ) (k − kc )3 . 6 ∂k 3

(32)

Generation of transient waves by impulsive disturbances in an inviscid fluid with an ice-cover

55

Thus, the stationary-phase approximation to Eq. (19) for x/t = C gmin is given as     1 kc 1 sin ψc , (1 + σ kc ) cos ψc , {ζ1 , ζ3 , ζ4 } × exp(kc z 0 ) cos ψc , 3 π(36|ωc |t)1/3 (1 + σ kc ) ωj (33) where π ψc = kc |x| − ωc t − sgn(x)sgn(ωc ) , 6

(34)

ωc = ω(kc , h) and (·) is the Gamma function. When |x|/t < C gmin , Eq. (22) has no real positive roots. However, as |x|/t lower than C gmin is sufficiently near to C gmin , we may expand the phase function near kc as follows:

mn (k) ≈ mn (kc ) +

∂ mn (kc ) 1 ∂ 3 mn (kc ) (k − kc )3 . (k − kc ) + ∂k 6 ∂k 3

(35)

Thus, Eq. (19) can be approximately expressed as {ζ1 , ζ3 , ζ4 }

1 (32ωc t)1/3 (1 + σ kc )    2  O O m ikc O O × Re exp(iψm ) exp(kc z 0 )Ki(Z m ), (−1) Ki(Z m ), (1 + σ kc )Ki(Z m ) , ωc

(36)

m=1

where ψmO = (−1)m+1 (kc |x| − ωc t), Z mO

= (−1)

m+1



|x| − ωc t



2 ωc t

(37)

1/3 ,

and ωc = ∂ω(kc )/∂k. As h tends to zero, Eq. (24) simply reduces to   2   g 1/2 t t gt z 0 , ζ , ζ } , , cos ψ {ζ1 3 4 exp cos ψ0 sin ψ0 √ 0 , 4x 2 2x 2 π|x|3/2

(38)

(39)

where ψ0 = −

gt 2 π + sgn(x) , 4x 4

(40)

which corresponds to the free-surface gravity waves obtained in the Cauchy-Poisson problems. Obviously, ζ3 and ζ4 in Eq. (39) are in accord with Stoker’s Eqs. (6.5.13) and (6.5.12), respectively [7, p. 166]. For the pure gravity waves, ∂ 2 ω0 /∂k 2 < 0 holds for all k > 0. 4 Formal solution and asymptotic representation for the three-dimensional problem For three-dimensional cases, the singular component in Eq. (8) is the fundamental solution of Laplace equation in an unbounded domain, M δ(t) 4πr c+i∞  +∞ +∞ 1 M ds dαdβ exp[−K |z − z 0 | + i F + st], =− 16π 3 i K


S = −

c−i∞

−∞ −∞

(41)

56

D.Q. Lu, S.Q. Dai

 where K = α 2 + β 2 , F = αx + βy. By taking a Laplace-Fourier transform similar to Eq. (14), we have the solutions for the three-dimensional problem: +∞ +∞ 1 exp(i F) {ζ1 , ζ3 , ζ4 } = dαdβ 4π 2 1+σK −∞ −∞   K × exp(K z 0 ) cos(t), − sin(t), (1 + σ K ) cos(t) , 

(42)

where (K , h) = ω(K , h). Obviously, the integrand in Eq. (42) is similar to that in Eq. (16) with t0 = 0. Furthermore, with a change of variables {x, y} = R{cos θ, sin θ}, {α, β} = K {cos φ, sin φ}, Equation (42) can be re-written as {ζ1 , ζ3 , ζ4 } =

 +∞ 1 K J0 (K R) dK 2π 0 1+σK   K × exp(K z 0 ) cos(t), − sin(t), (1 + σ K ) cos(t) , 

(43)

where J0 (K R) is the zeroth-order Bessel function of the first kind. It is noted that ζ4 agrees with Maiti and Mandal’s Eq. (23) [6] for the deflexion due to a displacement concentrated at the origin. For the sake of consistency, we replace K and  with k and ω respectively for Eq. (43) without loss of its exactness. Furthermore, we may replace J0 (k R) by its asymptotic formula for large k R [18, p. 364],  1/2  2 π . (44) J0 (k R) cos k R − πk R 4 Thus, we have an approximation for Eq. (43) as follows +∞ 1/2   2 2  k exp(itmn ) 1  n+1 ik {ζ1 , ζ3 , ζ4 } dk × exp(kz 0 ), (−1) ,1 + σk , 4π 2π R 1 + σk ω

(45)

m=1 n=1 0

where π (−1)m+1  kR − + (−1)n+1 ω. (46) t 4 It is evident that Eqs. (45) and (46) are similar to Eqs. (19) and (20), respectively. Therefore, the asymptotic analysis for Eq. (45) follows that in Sec. III. The stationary points for mn can be determined by Q(k, K 0 , h) = 0 with K 0 = gt 2 /4R 2 . The procedure will not be repeated here and the final results are simply quoted as follows. When R/t > C gmin , mn =

{ζ1 , ζ3 , ζ4 }

2  j=1

 kj sin ϕ j , (1 + σ k j ) cos ϕ j , (47) × exp(k j z 0 ) cos ϕ j , 2π(R|ωj |t)1/2 (1 + σ k j ) ωj 

1/2

kj

where

π ϕ j = k j R − ω j t − [1 + sgn(ωj )] . 4 When R/t > C gmin and is sufficiently close to C gmin , {ζ1 , ζ3 , ζ4 }

2  2 

(48)

1/2

kj

1/3 (1 + σ k ) (2π R)1/2 (32ω j j t)    ik j Ki(Z Ij ), (1 + σ k j )Ki(Z Ij ) , ×Re exp(iϕmI j ) exp(k j z 0 )Ki(Z Ij ), (−1)m ωj m=1 j=1

(49)

Generation of transient waves by impulsive disturbances in an inviscid fluid with an ice-cover

where

 ϕmI j = (−1)m+1

(ωj )3 t π kj R − − ωjt − , 2 4 3(ω j )

57

(50)

and Z Ij is given by Eq. (30). When R/t = C gmin , {ζ1 , ζ3 , ζ4 }

    1/2 1 kc kc z ) cos ϕ , sin ϕ , (1 + σ k ) cos ϕ × exp(k c 0 c c c c , 3 (2π 3 R)1/2 (36|ωc |t)1/3 (1 + σ kc ) ωc (51)

where ϕc = kc R − ωc t −

π π − sgn(ωc ) . 4 6

(52)

When R/t < C gmin and is sufficiently close to C gmin , 1/2

{ζ1 , ζ3 , ζ4 }

kc 1/2 (2π R) (32ωc t)1/3 (1 + σ kc )   2  ikc × Re exp(iϕmO ){exp(kc z 0 )Ki(X mO ), (−1)m Ki(X mO ), (1 + σ kc )Ki(X mO )} , (53) ωc m=1

where

 π ϕmO = (−1)m+1 kc R − ωc t − , 4 X mO

= (−1)

m+1



R

− ωc t



2 ωc t

(54)

1/3 .

As h tends to zero, Eq. (47) simply reduces to   2   gt 2 t gt z 0 {ζ1 , ζ3 , ζ4 } 5/2 3 exp , , cos ϕ cos ϕ sin ϕ 0 0 0 , 4R 2 2R 2 πR

(55)

(56)

where ϕ0 = −

gt 2 , 4R

(57)

which corresponds to the pure gravity waves in the Cauchy-Poisson problems. ζ3 in Eq. (56) is in accord with Stoker’s Eq. (6.5.15) [7, p. 166]. 5 Discussion and conclusion The general characteristics of the Cauchy-Poisson wave motions have been in detail discussed by Stokers [7, Sect. 6.6], Debnath [9; 19, Sects. 3.2, 3.3] and Mei [14, Sect. 2.1]. The dispersive characteristics and wave systems for the case of three-dimensional problem are of the same nature as those for the two dimensions since (K , h) and ω(k, h) have the same analytic form. The oscillatory factors in two- and three dimensions do not differ essentially from each other, while the wave amplitude factors are different. Waves in two dimensions are symmetric with respect to the z axis while waves in three dimensions have cylindrical symmetry. The principal interest here is to find the distinct effect of the presence of an ice-cover on the wave motion, which was first considered by Maiti and Mandal [5,6] but has not been explored fully. Next, the characteristics of the two-dimensional symmetric wave motions are discussed and the graphical representations are provided for the case of x > 0. One can find that the conclusions reached here are also applicable for the axisymmetric waves.

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D.Q. Lu, S.Q. Dai

According to Stoker’s theory [7, p. 169], the definitions for the local wave length and wave period can be given as  {λ j , T j , C pj , C g j } =

 2π 2π ω j  , , , ω j , ( j = 0, 1, 2). kj ωj kj

(58)

Obviously, for a fixed h, λ1 − λ2 > 0 and T1 − T2 > 0 hold invariably since k1 < k2 and ω(k, h) is a monotonously increasing function with respect to k. λ1 and T1 increase sightly with increasing h while λ2 and T2 increase sharply. It is well known that for a fixed x, λ0 (x/t) = 8π x 2 /gt 2 and T0 (x/t) = 4π x/gt decrease as the time increases, while for a fixed t, λ0 and T0 increase with increasing x [7, p. 163]. It can be seen from Figs. 2 and 3 that the behaviors of λ1 (x/t, h) and T1 (x/t, h) are similar to those of λ0 and T0 , respectively. However, the behaviors of λ2 (x/t, h) and T2 (x/t, h) are on the contrary. It is notable that for a fixed x, λ2 and T2 increase as the time increases, while for a fixed t, λ2 and T2 decrease with increasing |x|. A close examination of Figs. 2 and 3 shows that there exist transition positions, denoted by xλ (t, h) and x T (t, h) respectively, for λ = λ1 − λ0 and T = T1 − T0 . Once t and h are specified, λ ≶ 0 when x ≶ xλ while T ≶ 0 when x ≶ x T , which are illustrated in Figs. 4 and 5. As it is, one may prove the existence of xλ and x T by an analysis on the general dispersion relation. Graphical representations in Figs. 6 and 7 show that there exist transition wave numbers, denoted by kt p (h) and ktg (h) respectively, for C p (k, h) = C p (k, h) −C p (k, 0) and C g (k, h) = C g (k, h) − C g (k, 0), where C p (k, h) = ω/k, C p (k, 0) and C g (k, 0) correspond respectively to (i)

20

(ii)

(iii) (iv)

Wavelengths

15 l0 l1 l2

10 (i) (ii) (iii) (iv)

5

0

0

50

100

x

Fig. 2 Wavelengths λ j (x/t, h) with h = 0.01 m at (i) t = 5 s, (ii) t = 10 s, (iii) t = 15 s, and (iv) t = 20 s (i)

5

(ii)

(iii)

(iv)

Wave periods

4 T0 T1 T2

3 (i) (ii) (iii) (iv)

2 1 0

0

50

x

100

Fig. 3 Wave periods T j (x/t, h) with h = 0.01 m at (i) t = 5 s, (ii) t = 10 s, (iii) t = 15 s, and (iv) t = 20 s

Generation of transient waves by impulsive disturbances in an inviscid fluid with an ice-cover

59

∆λ

0

-0.5

(i) (ii)

(iii) (iv)

0

50

100

150

x Fig. 4 λ = λ1 − λ0 with h = 0.01 m at (i) t = 5 s, (ii) t = 10 s, (iii) t = 15 s, and (iv) t = 20 s

∆T

0

-0.05 (i) (ii) (iii)

-0.1 0

(iv)

50

x

100

150

Fig. 5 T = T1 − T0 with h = 0.01 m at (i) t = 5 s, (ii) t = 10 s, (iii) t = 15 s, and (iv) t = 20 s (vi) (v)

(iv)

(iii)

∆C p

0.025

(ii) (i)

0

-0.025

0

1

2

3

k Fig. 6 C p = C p (k, h) − C p (k, 0) with (i) h = 0 m, (ii) h = 0.001 m, (iii) h = 0.005 m, (iv) h = 0.01 m, (v) h = 0.05 m, and (vi) h = 0.1 m

the phase speed and the group velocity for the free-surface gravity waves in an inviscid fluid of infinite depth. For a given h, C p ≶ 0 when k ≶ kt p while C g ≶ 0 when k ≶ ktg . Generally, xλ = x T and kt p = ktg . Physically, an observer moving at a certain speed x/t > C gmin will see two trains of waves, namely long gravity waves with wave number k1 and short flexural waves with wave number k2 . Moreover, observers with different moving speeds will see different wave profiles. Figure 8 shows the deflexions of the ice-water interface for different observers moving with a speed larger than, larger than but close to, at, lower than but close to, the minimal group velocity, C gmin (0.01) ≈ 1.91007m/s. An observer moving at a certain speed much lower than

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D.Q. Lu, S.Q. Dai

(vi) (v) (iv)

(iii)

0.025

∆Cg

(ii)

(i)

0

-0.025

0

1

2

3

k Fig. 7 C g = C g (k, h) − C g (k, 0) with (i) h = 0 m, (ii) h = 0.001 m, (iii) h = 0.005 m, (iv) h = 0.01 m, (v) h = 0.05 m, and (vi) h = 0.1 m 0.15

(i) (ii) (iii) (iv)

0.1

z

0.05 0 -0.05 -0.1 5

10

15

20

25

t Fig. 8 Evolution of ζ1 (x/t, h) with h = 0.01 m and z 0 = −1.0 m for an observer with a fixed moving speed, (i) x/t = 2.0 m/s; (ii) x/t = 1.91017 m/s; (iii) x/t = C gmin ≈ 1.91007 m/s; (iv) x/t = 1.909 m/s

Ice-covered waves Free-surface waves

z

0.05

0

-0.05

50

100

150

200

x Fig. 9 Wave profile ζ1 (x/t, h) with h = 0.01 m and z 0 = −1.0 m at t = 15 s for observers with x/t > C gmin

C gmin will see no wave motion at the observation point. Figures 9–14 show the wave profiles obtained by summing up the views of many observers with x/t > C gmin for the same instant, where C gmin (0.05) ≈ 3.47434m/s. It can be seen that there are two spatial scales in the ice-covered wave motions. One is long gravity waves, and the other the short flexural waves. The long waves resemble those for the free-surface gravity waves while the

Generation of transient waves by impulsive disturbances in an inviscid fluid with an ice-cover

61

Ice-covered waves Free-surface waves

0.05

z

0

-0.05

50

100

150

200

x Fig. 10 Wave profile ζ1 (x/t, h) with h = 0.05 m and z 0 = −1.0 m at t = 15 s for observers with x/t > C gmin Ice-covered waves Free-surface waves

0.05

z

0

-0.05

50

100

150

200

x Fig. 11 Wave profile ζ3 (x/t, h) with h = 0.01 m at t = 15 s for observers with x/t > C gmin Ice-covered waves Free-surface waves

z

0.05

0

-0.05 50

100

150

200

x Fig. 12 Wave profile ζ3 (x/t, h) with h = 0.05 m at t = 15 s for observers with x/t > C gmin

short waves ride on the long waves in the course of propagation. It can be seen from Figs. 9–14 that the effects of the thickness of ice-plate on the wave profiles, wavelengths, and wave periods are significant. Obviously, as h increases, the wave amplitudes of ζ12 increase due to the decreasing submergence decay factors while those of ζ22 and ζ32 decrease, and at the same time the wavelengths and periods of all waves increase. As the first step to solve the full problem which involves general disturbances, the asymptotic solutions for the waves due to fundamental singularities have been derived in this paper. It should be noted that the analytical solutions obtained here are valid for large time with a fixed distance-to-time ratio. Consequently, the solutions

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D.Q. Lu, S.Q. Dai

Ice-covered waves Free-surface waves

z

0.1

0

-0.1

50

100

x

150

200

Fig. 13 Wave profile ζ4 (x/t, h) with h = 0.01 m at t = 15 s for observers with x/t > C gmin Ice-covered waves Free-surface waves

z

0.1

0

-0.1

50

100

150

200

x Fig. 14 Wave profile ζ4 (x/t, h) with h = 0.05 m at t = 15 s for observers with x/t > C gmin

obtained here describe the views of observers moving with local group velocity. For an observer moving with fixed x/t, the wave amplitudes will decay as the waves spread to infinity. By use of the formulae presented by Mei [14, p. 29], one can demonstrate that the total energy of the waves between two observers moving with the local group is conserved. However, the asymptotic solutions show that the amplitude at a fixed point x is seem to grow large without limit as t tends to infinity. One possible reason for this singular behavior, as discussed by Stoker [7, p. 167], lies in the strong singularity at the source point. Once the finite disturbances are applied, the wave amplitudes would remain bounded with increasing t. Recently, Chen and Duan [10] demonstrated that the nonphysical behavior for Cauchy-Poisson gravity waves due to a simple source in an inviscid fluid without an ice-cover can be eliminated by the inclusion of the surface tension while Lu and Chwang [11] proposed another approach by introducing the viscosity of the fluid. The effects of surface tension and viscosity on the waves generated in an ice-covered fluid remain unsolved mathematical tasks and will be studied in the future. Acknowledgements This research was sponsored by the State Key Laboratory of Ocean Engineering, China, under Grant No. 0502 and the Shanghai Leading Academic Discipline Project under Project No. Y0103.

References 1. Squire, V.A., Hosking, R.J., Kerr, A.D., Langhorne, P.S.: Moving Loads on Ice Plates. Kluwer Academic Publishers, the Netherlands (1996) 2. Miles, J., Sneyd, A.D.: The response of a floating ice sheet to an acceleration line load. J Fluid Mech 497, 435–439 (2003) 3. Chowdhury, R.G., Mandal, B.N.: Motion due to ring source in ice-covered water. Int J Eng Sci 42, 1645–1654 (2004) 4. Mandal, B.N., Basu, U.: Water diffraction by a small elevaltion of the bottom of an ocean with an ice-cover. Arch Appl Mech 73, 812–822 (2004) 5. Maiti, P., Mandal, B.N.: Water waves generated by disturbances at an ice cover. Int J Math Math Sci 2005, 737–746 (2005)

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6. Maiti, P., Mandal, B.N.: Water waves generated due to initial axisymmetric disturbance in water with an ice-cover. Arch Appl Mech 74, 629–636 (2005) 7. Stoker, J.J.: Water waves: the mathematical theory with applications. Interscience Publishers, New York (1957) 8. Miles, J.W.: The Cauchy-Poisson problem for a viscous liquid. J Fluid Mech 34, 359–370 (1968) 9. Debnath L.: The linear and nonlinear Cauchy-Poisson wave problems for an inviscid or viscous liquid. In: Rassias, T.M. (ed) Topics in mathematical analysis, World Scientific, Singapore, pp 123–155 (1989) 10. Chen, X.B., Duan, W.Y.: Capillary-gravity waves due to an impulsive disturbance. In: Proceedings 18th international workshop on water waves and floating bodies, Le Croisic, France, 4 pp. (available online at http://www.rina.org.uk/index.pl?section=IWWWFB) (2003) 11. Lu, D.Q., Chwang A.T.: Free-surface waves due to an unsteady stokeslet in a viscous fluid of infinite depth. In: Cheng, L., Yeow, K. (eds.) Proceedings the 6th international conference on hydrodynamics, Perth, Western Australia, Taylor & Francis Group, London, pp 611–617 (2004) 12. Lu, D.Q., Wei, G., You, Y.X.: Unsteady interfacial waves due to singularities in two semi-infinite inviscid fluids. J Hydrodynam B 17, 730–736 (2005) 13. Miles, J.W.: Transient gravity wave response to an oscillating pressure. J Fluid Mech 13, 145–150 (1962) 14. Mei, C.C.: The applied dynamics of ocean surface waves. World Scientific Publishing, Singapore (1994) 15. Nayfeh, A.H.: Introduction to perturbation techniques. Wiley-Interscience, New York (1981) x 16. Scorer, R.S.: Numerical evalution of integrals of the form I = x12 f (x) exp(iφ(x))dx and the tabulation of the function  +∞ Gi(z) = (1/π) 0 sin(uz + 13 u 3 )du. Q J Mech Appl Math 3, 107–112 (1950) 17. Vallée, O., Soares, M.: Airy functions and applications to physics. Imperial College Press, London (2004) 18. Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards, Washington D.C. (1972) 19. Debnath, L.: Nonlinear water waves. Academic, Boston (1994)